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Introduction

This paper examines the contribution of ability to the rise in the economic

return to education. A common view in both the popular and professional

literatures is that much of the increase in the return to education can be

attributed to an increase in the return to ability. Herrnstein and Murray

(1994) make this a cornerstone of their analysis. They refer to the research

of Blackburn and Neumark (1993) who report that the rise in the economic

return to education is concentrated among those with high ability, a di¤erent

proposition from the one stated by Herrnstein and Murray, but not necessar-

ily inconsistent with it. In a similar vein, Murnane, Levy and Willett (1995)

conclude that a substantial fraction of the rise in the return to education

between 1978 and 1986 for young workers can be attributed to a rise in the

return to ability. When they condition on ability, the rise in the economic

return to education is diminished.

The implicit assumptions that govern much of this literature are (1) that

ability is valued in the market (or is a proxy for characteristics that are val-

ued), (2) that the price of ability (or the proxied characteristics) is rising in

the new market for skills, and (3) that ability is correlated with education.

As a consequence of these assumptions, failure to control for ability leads

to an upward bias in the estimated economic return to education, and the

bias is greater in periods when the return to ability is greater. This is one

possible explanation for a positive interaction of education, time and ability.

Other explanations are (a) that the correlation between ability and schooling

3

is increasing over time, due to increasing application of the meritocratic prin-

ciple in educational enrollment, even if the return to ability remains constant

(Herrnstein and Murray, 1994) or (b) that ability-education bundles produce

skills that are more valued in the new economy, the skills are superadditive

functions of ability and education (Rubinstein and Tsiddon, 1999) and the

demand for the highest skills has increased disproportionately.

The small ability bias reported in Chamberlain and Griliches (1975) may

be a consequence of the low economic return to ability in the time period

of their samples. Ability bias will be greater in an era with greater return

to ability or a more meritocratic relationship between schooling and ability.

Herrnstein and Murray (1994) argue that both of these factors are at work

in the modern economy.

Ability bias is usually discussed as a problem of omitted variables (see,

e.g. Griliches, 1977 or Chamberlain and Griliches 1975). Include the missing

ability variable and, except for problems of measurement error, there will be

no bias. The conventional formulation of the ability bias problem ignores

the strong dependence between education and ability which Herrnstein and

Murray (1994) argue has become stronger in recent years. If the dependence

between ability and education becomes too strong, it is impossible to isolate

the e¤ect of education from ability even when the latter is perfectly observed.

This gives rise to the logically prior problem of sorting bias, discussed in this

paper.

Table 1 shows that there are very few white male college graduates with

low ability in the NLSY. Further, there are no white men with postgradu-

4

ate education in the lowest ability quartile, so for that ability quartile, no

estimate of the wage gain of such education is possible. For many schooling-

ability pairs, the cells are empty (or nearly so), making it di¢cult to isolate

separate ability e¤ects and schooling e¤ects, and making main e¤ects of abil-

ity and education di¢cult, if not impossible, to identify. In the limit, if ability

and education are perfectly strati…ed, there is no way to isolate returns to

education from returns to ability, even if ability is perfectly measured. Em-

pirically, the two are indistinguishable.1

Missing data also complicate attempts to separate the e¤ects of age and

time. Estimates of the role of ability in explaining the increasing return to

schooling that are reported in the recent literature follow the same people,

or repeated cross section samples of the same cohorts, over time. To follow

the same people or cohort over time is also to follow them as they age. The

econometric problem created by such samples is more severe than the usual

age-period-cohort e¤ect problem.2 Figure 1 is a Lexis diagram for a single

cohort of a speci…ed initial age followed over time. Darkened cells indicate the

data that exist for each age and time period. If panel data or repeated cross

section data consist of only a single age cohort, age and time are hopelessly

confounded. It is impossible to identify separate age and time e¤ects. Even

with multiple age cohorts (see, e.g., Figure 2 for the data structure of the

NLSY panel) there are many empty cells. The “main e¤ects” for time or

age, de…ned as averages over entire rows and columns, cannot be computed.

(In the age-period-cohort problem these averages can be identi…ed if cohort

e¤ects are suppressed.) Some of the components required to form these

5

means are missing. It is also impossible to identify interactions associated

with the empty cells without imposing parametric structure (e.g., that age

and time e¤ects are linear so that trends …t on nonempty data cells apply to

the empty ones).

The current literature on ability bias ignores the …rst problem (strong

dependence between education and ability) and implicitly solves the second

problem in two distinct ways. Some authors impose linearity of time and/or

age e¤ects (e.g. Blackburn and Neumark, 1993; Bishop, 1991; Grogger and

Eide, 1995) and arbitrarily suppress certain interactions.3 Although a fully

nonparametric model is not identi…ed, the hypothesis of linearity is testable.

We demonstrate that the NLSY data are at odds with the widely-used as-

sumptions that time and age e¤ects are linear. Invoking linearity solves the

identi…cation problem but imposes unjusti…ed restrictions across time peri-

ods and ages. Murnane, Levy and Willet (1995) solve the second problem in

a di¤erent way by estimating the contribution of ability to eliminating the

rise in the return to education measured at one age in two di¤erent years.

This procedure leaves open the question of whether their results are special

to the age they choose.

This paper is organized into two sections. The …rst section discusses the

identi…cation problems that arise from using panel data or repeated cross

section data to separate time and age e¤ects that arise from the strong strat-

i…cation of ability and education. There we present the combinations of

parameters that can be estimated from panel data. An appendix derives

the precise combinations of interactions that can be identi…ed when cells are

6

missing.

In Section 2 we test and reject the widely-used speci…cation that age and

time e¤ects are linear. Estimates from nonparametric procedures indicate

mild support for the point of view that in the mid-80s there was an increase

in the college-high school wage di¤erential for the most able. This pattern

is not found for other ability and schooling groups for which nonparametric

estimates can be obtained. This produces a more nuanced interpretation of

the ability - schooling - time interaction than that reported in the recent

literature.

1 Estimating Interactions and Main E¤ects

From Incomplete Data

Assume that the log wage at age a and time t can be decomposed into

main e¤ects and interaction:

`n w(a; t) = ®(a) + ¯(t) + °(a; t); a = 1; ::; A; t = 1; ::; T

where ®(a) is the age main e¤ect, ¯(t) is the time main e¤ect and °(a; t)

is the interaction of age and time. To simplify the exposition, we implicitly

condition on education and ability.

The bene…t of observing two age cohorts facing common year e¤ects is

that we observe the same age in two di¤erent years (except for certain ages

in the …rst and last years), and two di¤erent ages in the same year. With

access to such data, we can estimate a nonparametric additive model

7

(1.1) `n w(a; t) = ®(a) + ¯(t); a = 1; ::; A; t = 1; ::; T

if we suppress the interaction °(a; t). First, we make one normalization, e.g.

®(1) = 0. With this normalization, ¯(1) is identi…ed. Using this knowledge

of ¯(1), we can identify ®(2) since

`n w(2; 1) = ®(2) + ¯(1):

Proceeding in this fashion, the main time and age e¤ects are identi…ed.4

It is also possible in this case to identify an interaction between age and

time if we assume, as does much of the literature, that age and time e¤ects

are linear i.e. if we assume that

¯(t) ¡ ¯(t¡ ¢) = b¢

®(a) ¡ ®(a¡ ') = c'

where b and c are scalars and ¢ and ' are integers. Under the assumption

of linearity, it is possible to identify interaction °(a; t) and hence term d

in °(a; t) = dat; provided that T ¸ 2 and A ¸ 2. There are only three

parameters, and they can be identi…ed from four or more cells.

It is important to observe that identi…cation is achieved by imposing

arbitrary conventions. In the NLSY, only the blackened cells in Figure 2

are available. The problem of empty o¤-diagonal cells substantially restricts

what can be learned in two major ways. First, it prevents identi…cation of

unconditional age and time main e¤ects. The unconditional time e¤ect is the

average time-speci…c e¤ect for every age cohort, not just those observed in the

8

data. Likewise, the unconditional age e¤ect is the average e¤ect for persons

of a given age, across all time periods, not simply those observed in the data.

Since we do not observe every age in every year (i.e., since we have empty

o¤-diagonal cells), it is impossible to estimate these unconditional e¤ects.

Instead, we can estimate conditional main e¤ects: age e¤ects conditional on

the times observed, and time e¤ects conditional on the ages observed. This

problem is distinct from the linear dependence that arises in the standard

age-period-cohort e¤ect problem. That problem arises even when all the cells

of Figures 1 and 2 are available. The problem discussed in this paper arises

even if there are no cohort e¤ects. A formal comparison of unconditional and

conditional e¤ects is presented in Appendix B.

The second major e¤ect of empty data cells is to limit the number of

identi…able interactions. Speci…cally, interactions associated with empty data

cells obviously cannot be identi…ed. If only one age cohort is observed (as in

Figure 1), no main e¤ects or interactions are identi…ed. They are hopelessly

confounded as the single age cohort simultaneously ages and enters a new

economic environment. Given two age cohorts, all main e¤ects are identi…ed

if interactions are assumed to be zero. For three or more age cohorts, certain

combinations of interactions are identi…ed. Individual interactions cannot be

identi…ed. The problem is more severe at the boundary ages (for the youngest

and oldest workers) where certain ages are observed for the …rst or last time.

This feature of the identi…cation problem is unfortunate because, as noted in

Cawley, Heckman, Lochner and Vytlacil (1999), considerable attention has

been devoted to interactions for the youngest age groups in the NLSY.

9

The absence of identi…able interactions outside the black band displayed

in Figure 2 means that any test for the absence of interactions is actually tests

whether or not linear combinations of the identi…ed interactions are zero. The

distinction is important because even if there are nonzero interactions, it is

possible that the combination of interactions that can be estimated will be

zero. Any test will have zero power against such an alternative.5 The com-

binations of interactions that can be identi…ed and tested are characterized

in Appendix B.

The literature copes with the identi…cation problem in various ways. Dif-

ferent strategies lead to very di¤erent empirical results. Bishop (1991) as-

sumes linear time and age e¤ects. Blackburn and Neumark (1993) assume

linear age e¤ects and linear time e¤ects in the interactions they estimate.

Grogger and Eide (1995) assume linear experience e¤ects and but no age

e¤ects. None of the studies summarized in Cawley, Heckman, Lochner and

Vytlacil (1999) …ts a model with time and age e¤ects estimated for each

education-ability cell. Studies di¤er in which interactions are estimated and

suppressed.

We have outlined the limitations that stem from empty data cells. How-

ever, there is an additional estimation problem that is tantamount in practice

to an identi…cation problem: data cells that are nonempty but contain little

data. The problem of missing data on age and time is compounded because

estimates are often conditional on ability and education, making the problem

one of missing and sparse data in a four-dimensional grid (age, time, ability,

and education). In addition, some ability-education cells are missing and

10

others are sparse (see Table 1). This means that it is impossible to identify

all education-ability interactions. Main e¤ects for education are formed only

over a subset of the ability cells. In the limit, with perfect strati…cation of

education with ability, the main e¤ects are interaction e¤ects. The inabil-

ity to identify main e¤ects attributable to either ability or education is the

problem of sorting bias.

The next section of the paper reexamines the wage returns to ability and

education. We nonparametrically estimate conditional time and age main

e¤ects and the identi…ed combinations of interactions. In order to conduct a

nonparametric analysis, we necessarily must limit the number of variables we

include in the model. This means that our models contain fewer regressors

than previous models that investigate the returns to ability over time and

the education-ability-time interaction.

2 Nonparametric Estimates of Main E¤ects

and Interactions

To address these identi…cation problems, we use extracts from the NLSY

data documented in Appendix A. The NLSY is a panel data set with unusu-

ally rich information on measures of cognitive ability. Table 2 presents the

components of the ASVAB test reported in the NLSY.

For our measure of ability, we use general intelligence, or g, which we

take as the …rst principal component of the ASVAB test scores.6 There

11

has been considerable debate about what represents the best measure of

cognitive ability. General intelligence, which re‡ects the ability to perform

well on the tests used to estimate it, is commonly used in psychometrics,

though it is often supplemented with more speci…c ability measures (see, e.g.

the review in Carroll 1997). In Cawley, Conneely, Heckman and Vytlacil

(1997) we show that there is little di¤erence between general intelligence,

(Armed Forces Qualifying Test), or averages of the ASVAB test of the sort

used by Blackburn and Neumark (1993), in terms of explanatory power in log

wage regressions. In parallel analyses of the sort we conduct in this paper,

using the measure employed by Blackburn and Neumark, and for each of the

ASVAB test scores separately, we …nd qualitatively similar results for each

measure with the exception of Paragraph Completion.7

We have already presented our evidence on sorting bias and it is sum-

marized in Table 1. Figure 3 shows that there was a rise in the return to

college education in the mid-80s for white males in the NLSY. However, as

Murnane, Levy and Willet (1995) claim, this may largely be a consequence

of a rise in the return to cognitive ability over time. Figure 4 suggests that

the wage gap between individuals in the upper and lower quartile of ability

rose over this period.

Many hypotheses are consistent with the data, including: a rising return

to education with age, a rising return to ability with age, a rising return

to education with work experience, and a rising return to ability with work

experience.

We address two questions. (1) Is the rising return to education concen-

12

trated among the most able? We investigate this question using a nonpara-

metric approach. We estimate time e¤ects within education-ability-age cells.

(2) The second question addressed in this paper is whether we need to be so

agnostic about the parameterization of time and age. We test whether the

assumption of linear trends in time and age is justi…ed, so that the simple

methods used in the previous literature can be vindicated. Unfortunately,

they cannot. Relaxing linearity substantially quali…es the interpretation of

interactions previously reported in the literature.

All of our analysis in this section is for white males. Sparse data within

cells prevent us from estimating our nonparametric models for all other

groups. We cannot pool these groups because, as we have shown elsewhere

(Cawley, Conneely, Heckman and Vytlacil, 1997), the wage returns to ability

and education di¤er signi…cantly across race and gender. A cost of adopting

a nonparametric approach is that we are forced to adopt a simpler model,

with fewer regressors, than has typically been estimated in this literature.

We use nonparametric methods to clarify the two stated questions. With a

data set the size of the NLSY, we cannot be fully nonparametric in using the

full array of variables presented in other studies in this literature.

2.1 Is the Return to Ability or Education Rising?

The …rst question we investigate in this section is whether the rising

return to education should be attributed to a rising return to ability. We

present empirical results for the case when ability is divided into quartiles

13

and education is broken down into three categories: high school dropout,

high school graduate, and college graduate. We de…ne these education lev-

els by highest grade completed less than 12, equal to 12, and equal to 16,

respectively. This results in twelve education-ability cells.8

Figure 5 plots the time trends from a speci…cation that does not allow for

age-time interactions. Within each education level, we run a spline regression

of log wage on ability with knots at the 25th, 50th, and 75th percentiles of

ability, where the coe¢cients of the spline regression are allowed to depend

on time and age in an additively separable manner. In particular, letting a

denote age, t denote time, e denote education level, c denote cognitive ability,

and qc denote quartile of cognitive ability, the speci…cation is:

(2.1) `n w =³®(a; e; qc) + ¯(t; e; qc)

´+

³°(a; e; qc) + ±(t; e; qc)

´c + ²

where ² is mean independent of the a; t; e and c, and where the regression

equation is constrained to be continuous in the cognitive ability score, c; and

linear in c within the ability quartiles.9 No functional form assumption is

imposed on the coe¢cients besides those required to constrain the equation

to be continuous in c for each age, time and education level. The coe¢cients

may vary with age, time, education, or ability quartile. The plotted point

estimates are …tted values with the ability level evaluated at the midpoint

of each ability quartile.10 The plotted con…dence bands are plus and minus

two standard errors, with the standard errors estimated by a robust Eicher-

White procedure allowing for correlation in log wages across time for a given

individual. Because of the strong association between ability and education,

estimates could only be obtained for high school dropouts in the bottom

14

two ability quartiles, high school graduates in all four quartiles, and college

graduates in the top two quartiles. The plots indicate falling wages for men

with less than a college education, and rising wages for college graduates in

the two highest ability quartiles.11

In addition to the additive speci…cation (2.1), we also control for age in

a di¤erent way. We estimate time coe¢cients within each age cell, which

permits interactions between age and time.12 In particular, we estimate the

following spline regression:

(2.2) `n w = ®(a; t; e; qc) + °(a; t; e; qc)c+ ²

where the regression equation is again constrained to be continuous in the

cognitive ability score, c, within quartiles and no functional form assumption

is imposed on how the coe¢cients vary with age, time or education.13 This

analysis is not without cost; by looking within smaller data cells, we obtain

noisy estimates.

From this analysis, we conclude that the wage premium for college grad-

uation (over high school graduation) rose in the mid-1980s for white males of

the highest g quartile in their mid-20s: Figure 6 presents the most interest-

ing of these estimated wage premia.14 Similar analysis, for the third quartile,

is reported in Figure 7. We …nd no increase in the wage premium for col-

lege graduation for those in the third quartile of ability, a result essentially

in agreement with the interaction of education, ability and time reported

in Blackburn and Neumark (1993). Their …nding of an interaction among

ability, education and time is supported but it is isolated in the highest g

quartile group. The e¤ect of ability on the education-time interaction is not

15

continuous. At lower ability or education levels, increases in ability do not

increase the education-time trend.

Figure 7 should be treated very cautiously due to small sample sizes.

There are more than twenty observations in each reported age-time cell for

fourth quartile college graduates and high school graduates, but there are less

than twenty observations in many reported age-time cells for third quartile

college graduates. Insu¢cient data prevent us from performing a parallel

analysis for the bottom two ability quartiles. For the high school graduate -

high school drop out wage di¤erential, there is little evidence of a rise in the

return to education for the ability cohorts where usable cells are available.15

Among the estimable cells, the rise in the wage di¤erentials among schooling

groups is only found among younger fourth quartile college graduates. In

a parallel analysis that controls for work experience instead of age, we …nd

a signi…cant time trend in the college graduate-high school graduate wage

di¤erential again in the mid 80s but only for workers with the least work

experience.16

2.2 Parameterizing Age and Time E¤ects

The nonparametric stance we take in this paper is very conservative. With

a little additional structure, a clearer story might emerge. The second ques-

tion considered in this section is whether we need to be fully nonparametric

in age and time. To answer this question we perform a series of tests.17

We …rst test whether time e¤ects are equal across education cells, in

16

particular, in the notation of equation (2.2) whether18

®(a; t+ 1; e; qc) ¡ ®(a; t; e; qc) = ®(a; t+ 1; e0; qc) ¡ ®(a; t; e0; qc)

°(a; t+ 1; e; qc) ¡ °(a; t; e; qc) = °(a; t+ 1; e0; qc) ¡ °(a; t; e0; qc)

for all available (a; t; qc; e), (a; t; qc; e0) cells with e 6= e0. We also test whether

time e¤ects are equal across ability quartiles, and whether age e¤ects are

equal across education and ability cells. We reject each of these four hy-

potheses, which implies that age and time e¤ects should be estimated within

education-ability cells.19

Next, within each education-ability cell, we test whether all identi…ed age-

time interactions are zero. In particular, we conduct a score test with the

unrestricted model given by equation (2.2) and the restricted model given

by equation (2.1). We reject the hypothesis of zero age-time interactions.

Combining the inferences from these tests, we conclude that in order to test

for the linearity of time e¤ects we must condition on age, and to test for

linearity of age e¤ects we must condition on time. We follow this strategy.

Speci…cally, for each age, we consider whether the age-speci…c time trend is

linear for each ability-education-age cell with data. The same approach is

used for testing whether the time-speci…c age trend is linear.20 We reject

the hypothesis that time e¤ects are linear across education-ability-age cells

and that age e¤ects are linear across education-ability-time cells. From this

entire series of tests, we conclude that there is no empirical justi…cation for

the widespread practice of assuming that the e¤ects of time and age are

17

linear.21

At the beginning of this section, we asked two questions. The …rst was:

how should attribution for the wage gain be divided between education and

ability? We have shown that education and cognitive ability are so strongly

associated that the wage e¤ects of the two cannot be separated for all groups.

This is a consequence of the problem of sorting bias previously discussed. We

…nd that the college graduate-high school graduate wage di¤erential rose in

the mid-80s for those in the highest quartile of ability but only for young

workers, (those with the least amount of work experience). High school

graduate-high school dropout wage di¤erentials are stagnant over time for

the lowest two quartiles of ability whether age or experience is used to control

for life cycle wage growth.

The second question asked was: do we need to be nonparametric when

estimating the e¤ects of age and time? The answer is yes. We …nd no

support for the widely accepted practice in the empirical literature of solving

the identi…cation problems posed in Section 1 by imposing linear e¤ects of

time and age. When this assumption is relaxed, we …nd that an education-

ability-time interaction only holds for high ability college graduates.

3 Conclusions

This paper examines the role of ability in accounting for the recent rise in

the economic return to education. Estimates of this e¤ect are often obtained

from panel data sets that follow a small range of birth cohorts over time.

18

The design of these data sets creates a serious identi…cation problem that

di¤erent authors cope with in di¤erent ways.22

In addition to the identi…cation problems raised by the panel structure

of the data used to isolate the e¤ect of ability, there is additional strati-

…cation of persons by ability into schooling strata. This gives rise to the

problem of sorting bias which is logically prior to the problem of ability bias

that has occupied the attention of empirical labor economists. If ability and

education are perfectly strati…ed, separate e¤ects of ability or schooling on

earnings cannot be identi…ed. With the levels of strati…cation in Table 1, sep-

arate ability and education e¤ects are estimable only by imposing arbitrary

parametric assumptions like linearity in age and education in an earnings

equation. In the literature, the ability bias problem is usually formulated as

a problem of omitted variables. The evidence reported in this paper suggests

that the real problem is that ability and schooling appear to be inseparable

— all interaction and no main e¤ects — even if ability is perfectly observed.

Sorting bias creates empty cells which compound the usual problems of iden-

tifying interactions. Di¤erent strategies for coping with these problems have

led to di¤erent interpretations of the role of ability in explaining the rising

return to schooling. It would be fruitful to conduct additional investigations

of sorting bias for data from earlier periods. Herrnstein and Murray (1994)

claim that strong sorting of ability and education is a recent phenomenon.

We show that a common method of “solving” the identi…cation problem,

assuming linear e¤ects of age and time, is not supported by the NLSY data.

We present nonparametric estimates of the identi…ed parameters in the data.

19

We …nd evidence that, within age groups, the college-high school premium

has increased in the mid 1980s for young persons of the fourth quartile of

ability but not for young persons in the third quartile of ability. Because of

the strong sorting of ability by schooling, the college-high school di¤erential

cannot be identi…ed for other quartiles and the estimated pattern is very

fragile for the third quartile of ability. When the strati…cation is made on

the basis of measured work experience, there is mild evidence of an increase

in the college-high school wage di¤erential for the most able men with low

levels of work experience. Few sturdy conclusions emerge about ability and

its e¤ect on the trend in the return to education for other groups.

20

References

[1] Bishop, John, “Achievement, Test Scores, and Relative Wages,” in

Kosters, Michael H. (editor), Workers and Their Wages: Changing Pat-

terns in the United States, (Washington, D.C.: American Enterprise

Institute Press, 1991).

[2] Blackburn, McKinley L., and David Neumark, “Omitted-Ability Bias

and the Increase in the Return to Schooling,” Journal of Labor Eco-

nomics, 11(3), (1993), 521-44.

[3] Carroll, John B. “Theoretical and Technical Issues in Identifying a Fac-

tor of General Intelligence,” in Bernie Devlin, Stephen Fienberg, Daniel

Resnick, and Kathryn Roeder (editors), Intelligence, Genes, and Suc-

cess: Scientists Respond to THE BELL CURVE. (Springer Verlag: New

York, 1997).

[4] Cawley, John, and Karen Conneely, James Heckman, and Edward Vyt-

lacil, “Cognitive Ability, Wages, and Meritocracy” in Bernie Devlin,

Stephen Fienberg, Daniel Resnick, and Kathryn Roeder (editors), Intel-

ligence, Genes, and Success: Scientists Respond to THE BELL CURVE,

(New York: Springer Verlag, 1997).

[5] Cawley, John, James Heckman, Lance Lochner, and Edward Vytlacil,

“Understanding the Role of Cognitive Ability in Accounting for the Re-

cent Rise in the Economic Return to Education,” in Kenneth Arrow,

21

Steven Bowles, and Steven Durlauf, (editors), Meritocracy and Eco-

nomic Inequality, (Princeton NJ: Princeton University Press, 1999).

[6] Chamberlain, Gary and Zvi Griliches, “Unobservables with a Variance

Components Structure: Ability Schooling and the Economic Success of

Brothers,” International Economic Review, 16(2), (1975), 422-449.

[7] Griliches, Zvi, “Estimating The Returns to Schooling: Some Economet-

ric Problems,” Econometrica, 45(1), (1977), 1-22.

[8] Grogger, Je¤ and Eric Eide, “Changes in College Skills and the Rise

in the College Wage Premium,” Journal of Human Resources, 30(2),

(1995), 280-310.

[9] Herrnstein, Richard J. and Charles Murray, The Bell Curve, (New York:

Free Press, 1994).

[10] Mason, William. and Stephen Fienberg, Cohort Analysis in Social Re-

search: Beyond the Identi…cation Problem, (New York: Springer Verlag,

1983).

[11] Murnane, Richard , John Willett, and Frank Levy, “The Growing Impor-

tance of Cognitive Skills in Wage Determination,” Review of Economics

and Statistics, 77(2), (1995), 251-266.

[12] Rubinstein, Yona and Daniel Tsiddon, “Coping with Technological

Progress: The Role of Ability in Making Inequality so Persistent,” Cen-

ter for Economic Policy Research Working Paper No. 2153, (1999).

22

[13] Searle, Shayle, Linear Models for Unbalanced Data, (New York: John

Wiley and Sons, 1987).

23

Notes1The evidence reported in Table 1 may be called into question because

education may increase ability. However, the level of sorting (as reported in

Table 1) is only slightly weaker if we consider only 14-16 year olds in 1979

in the NLSY whose ability is measured before they complete their schooling.

This table is available on request from the authors.

2See the essays in Mason and Fienberg (1983) for discussions of the clas-

sical age-period-cohort e¤ect problem.

3Cawley, Heckman, Lochner and Vytlacil (1999) summarize the literature

and demonstrate the sensitivity of estimates of ability-education-time e¤ects

to exclusion and inclusion of other variables, and suppression of certain in-

teractions.

4A similar identi…cation strategy entails normalizing ¯(1) = 0, and sub-

sequently identifying ®(1) and the rest of the main e¤ects.

5See e.g. Searle (1987).

6Because age at the time of test in‡uences test performance, we standard-

ize each of the ASVAB subtests to mean zero and variance one by age. We

calculate g as the …rst principal component of the standardized test scores.

For a more complete description of our measure of g and its characteristics,

see Cawley, Conneely, Heckman and Vytlacil (1997).

7When using Paragraph Completion as the measure of ability, we found

24

the time trends in the return to education to be qualitatively the same in the

third and fourth quartiles and could not reject the hypothesis that the time

trends were the same.

8We choose these divisions because they achieve a balance between dif-

ferentiating ability and education groups while still retaining enough obser-

vations in each cell to generate meaningful estimates.

9Experimentation with higher order splines produced similar though nois-

ier empirical results.

10Similar results are obtained using medians within quartiles.

11Our estimate of rising wages college educated individuals in the third

quartile of ability is fragile to the speci…cation used. The rising wage in

the third quartile for college graduates is not found with the alternative

speci…cation which conditions on ability quartile instead of using the linear

spline speci…cation. These results are available from the authors on request.

12The e¤ects of these two methods of “controlling” for a variable are often

confused in the literature, but only under the null hypothesis of no interac-

tions between age and time are the two methods equivalent.

13Use of higher order splines within ability quartiles does not a¤ect the

estimates.

14A full set of results is available from the authors upon request.

25

15Parallel analyses comparing the some college - high school graduate wage

di¤erential shows no rise in the wage di¤erential for the ability cohorts where

usable cells are available.

16These graphs are available from the authors upon request.

17We chose a signi…cance level of 1% for our hypothesis tests in this section.

Tables of p-values associated with all hypotheses tested in this section are

available upon request. We use a robust Eicher-White procedure for all tests.

18For the test of equality of age and time trends across education and

ability cells, we estimate equation (2.2) unrestricted and run a Wald test of

the given linear restrictions on the model.

19Details of these tests are available on request from the authors.

20For the linearity tests, we estimate equation (2.2) unrestricted and run

a Wald test of the appropriate linear restrictions on the model.

21Details of these tests are available from the authors on request.

22Cawley, Heckman, Lochner and Vytlacil (1999) demonstrate that small

changes in conventional speci…cations (adding and suppressing interactions

based on linear measures of age and time) produce very di¤erent estimates

of age - period - education e¤ects on wages.

26

Appendix A: Data

This paper uses the data from the National Longitudinal Survey of Youth

(NLSY). The NLSY, designed to represent the entire population of American

youth, consists of a randomly chosen sample of 6,111 U.S. civilian youths,

a supplemental sample of 5,295 randomly chosen minority and economically

disadvantaged civilian youths, and a sample of 1,280 youths on active duty

in the military. All youths were between fourteen and twenty-two years of

age when the …rst of annual interviews was conducted in 1979. The data

set includes equal numbers of males and females. 16% of respondents are

Hispanic and 25% are black. For our analysis, we restricted the sample to

those not currently enrolled in school and receiving an hourly wage between

$.50 and $1000 in 1990 dollars (all results of this paper are reported in 1990

dollars). Parallel analysis using $1 and $100 as the cut-o¤ points resulted in

similar results. This paper uses the NLSY weights for each year to produce

a nationally representative sample. However, our sample is not nationally

representative in age; we only observe a nine year range of ages in any given

year, and the oldest person in our 1994 sample is only 37.

In 1980, NLSY respondents were administered a battery of ten intelli-

gence tests referred to as the Armed Services Vocational Aptitude Battery

(ASVAB). Table 2 lists the ten tests. See Cawley, Conneely, Heckman and

Vytlacil (1997) for a more complete description.

27

Appendix B: Identifying Interactions In Incomplete Data

This appendix presents a formal analysis of identi…cation of interactions

when there are missing cells. First, we de…ne unconditional and conditional

main time and age e¤ects. Second, we describe the identi…ed combinations

of interactions in the presence of incomplete data with a pattern illustrated

in Figure 2. Assume A age groups and T time periods.

The problem of empty o¤-diagonal cells restricts what can be learned in

two major ways. First, it prevents identi…cation of unconditional main time

and age e¤ects. Let E(`n w(a; t)) = ¹(a; t). Unconditional main e¤ects are

de…ned as

®(a) = 1T

PTt=1 ¹(a; t); a = 1; :::; A

and

¯(t) = 1A

PAa=1 ¹(a; t); t = 1; :::; T .

Since we lack the data for every time and age, which are required to form

these sums, we cannot identify these parameters.

Without invoking further assumptions, we can only identify main time

e¤ects conditional on the ages observed. Assume that ¹A ages are observed

in each time period t, i.e. there are ¹A cohorts in the panel. For any t, the

youngest and oldest ages observed in any year are Af (t) = t and A`(t) =

t+ ¹A¡ 1.

The conditional main time e¤ect is:

28

¯(t; Af(t); A`(t)) = 1¹A

PA`(t)a=Af (t)

¹(a; t)

equivalently,

¯(t; Af (t); A`(t)) = ¯(t) + 1¹A

PA`(t)a=Af (t)

(®(a) + °(a; t)).

Estimated time e¤ects obtained by summing over available ages depend on

the interactions over the interval [Af(t); A`(t)].

Similarly, without making further assumptions, we can only identify the

main age e¤ect conditional on times observed. Let Tf(a) and T`(a) represent

the …rst and last years that age a is sampled. The conditional main age e¤ect

is:

®(a; Tf (a); T`(a)) =1

1 + T`(a) ¡ Tf(a)

T`(a)X

t=Tf (a)

¹(a; t)

equivalently,

®(a; Tf(a); T`(a)) = ®(a) +1

1 + T`(a) ¡ Tf(a)

T`(a)X

t=Tf (a)

(¯(t) + °(a; t)):

Estimated age e¤ects obtained by summing over available times depend on

the interactions over the interval [Tf(a); T`(a)].

Tf (a) and T`(a) can easily be related to the other parameters. Let T

equal the latest date in the panel which is also the oldest age. If every birth

cohort in the panel is observed passing through age a (i.e. ¹A · a · T ), then

age a is in the interior of the panel and Tf(a) = a¡ ( ¹A¡ 1) and T`(a) = a.

If not every birth cohort in the panel is observed passing through age a,

then age a is on the border of the panel. This is the case if a < ¹A or if

29

a > T . For ages on the border of the panel, Tf(a) = maxf1; a ¡ ( ¹A ¡ 1)gand T`(a) = minfa; Tg.

The second major e¤ect of empty data cells is to limit the number of

identi…able interactions. In a complete table, T (T +( ¹A¡1)) cells are de…ned

but only ¹AT are observed. For each t, only the cells (t; a = t); :::; (t; a =

t + ¹A ¡ 1) on or near the diagonal are observed in the panel structure.

In principle, no interaction for a (t; a) pair with width jt¡ aj > ¹A can be

nonparametrically identi…ed; i.e. only interactions associated with nonempty

data cells can be identi…ed. If only one age cohort is observed (i.e. ¹A =

1, as in Figure 1), no main e¤ects or interactions are identi…ed; they are

hopelessly confounded as the single age cohort simultaneously ages and enters

a new economic environment. For ¹A = 2, all main e¤ects are identi…ed if

all interactions are assumed to be zero. For ¹A ¸ 3, certain combinations of

the interactions are identi…ed without assuming zero interactions. Individual

interactions cannot be identi…ed.

The absence of identi…able interactions outside the blackened band dis-

played in Figure 2 means that any test for the absence of interactions is

actually a test that linear combinations of the identi…ed interactions are

zero. More precisely, we can always identify the combination of interactions

[°(a; t) ¡ °(a; t0)] ¡ [°(a0; t) ¡ °(a0; t0)]

for the set of all pairs ((t; a); (t0; a0)) 2 f(t; a); (t0; a0) j ` · a; a0 · ` + ¹A; for

` = t; t0; t; t0 = 1; :::; Tg. The di¤erence within brackets removes the common

30

additive age e¤ect and the di¤erence in di¤erences removes the common

additive time e¤ect. One can then test whether the residuals for the set of

all pairs ((t; a); (t0; a0)) jointly equal zero.

31

Table 1 -- Percent of Highest Grade Completed by Ability QuartileAge 30, White Males

Number of Observations: 1621

Highest Grade Completed Quartile 1 Quartile 2 Quartile 3 Quartile 4

7 2.0 0.0 0.0 0.0

8 6.7 0.5 0.0 0.0

9 10.4 1.7 0.2 0.0

10 7.9 3.2 0.0 0.0

11 9.6 2.7 1.0 0.0

12 54.0 63.2 46.9 22.5

13 3.9 7.2 11.1 4.4

14 3.0 7.9 10.1 10.6

15 0.5 1.7 3.9 4.9

16 2.2 9.6 1937 33.6

17 0.0 1.0 1.7 5.2

18 0.0 0.5 3.0 8.4

19 0.0 0.5 1.2 5.4

20 0.0 0.2 1.0 4.9Notes:1) Here, ability is defined as general intelligence, or ‘g’. We compute ‘g’ as the ASVAB test score vector times theeigenvector associated with the largest eigenvalue in the test score covariance matrix.2) Sample includes all respondents who were employed, out-of-school, and had valid observations each year fromage 24 to age 30. Anyone receiving more schooling after age 30 was excluded.

Table 2: The Armed Services Vocational Aptitude Battery

Subtest Minutes Description (A subtest of ASVAB measuring...)

General Science 11 Knowledge measuring the physical and biologicalsciences.

Arithmetic Reasoning 36 Ability to solve arithmetic word problems.

Word Knowledge 11 Ability to select the correct meaning of wordspresented in context and to identify the best synonymfor a given word.

Paragraph Comprehension 13 Ability to obtain information from written passages.

Numerical Operations 3 Ability to perform arithmetic computations (speeded).

Coding Speed 7 Ability to use a key in assigning code numbers towords (speeded).

Auto and Shop Information 11 Knowledge of automobiles, tools, and shopterminology and practices.

Mathematics Knowledge 24 Knowledge of high school mathematics principles.

Mechanical Comprehension 19 Knowledge of mechanical and physical principles andability to visualize how illustrated objects work.

Electronics Information 9 Knowledge of electricity and electronics.

ASVAB Testing Time 144

Figure 1: Lexis Diagram With a Single Age Cohort

Age (a)

Year(t)

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

Fig

ure

2: L

exis

Dia

gram

of

NL

SY P

anel

197

9-94

Age

(a)

Yea

r (t

)

1516

1718

1920

2122

2324

2526

2728

2930

3132

3334

3536

37

1979

XX

XX

XX

XX

1980

XX

XX

XX

XX

1981

XX

XX

XX

XX

1982

XX

XX

XX

X

1983

XX

XX

XX

XX

1984

XX

XX

XX

XX

1985

XX

XX

XX

XX

1986

XX

XX

XX

XX

1987

XX

XX

XX

XX

1988

XX

XX

XX

XX

1989

XX

XX

XX

XX

1990

XX

XX

XX

XX

1991

XX

XX

XX

XX

1992

XX

XX

XX

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1993

1994

XX

XX

XX

X

Yea

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